
where the choices of \(\hat{f}\) , \(\hat{u}^h\) , \(\widetilde{u}^h_x\)  and \(\check{u}^h_{xx}\)  are the same as in (REF )-(REF ). Here, \({\bf V}\)  is a subspace of \({\bf V}_N^k\)  which dynamically evolves over time [1]}. The adaptive procedure follows the technique developed in [1]} to determine the UWDG function space that dynamically evolves over time. The main idea is that in light of the distinguished property of the orthonormal multiwavelets, we keep track of multiwavelet coefficients as a natural error indicator for refining and coarsening, aiming to efficiently capture the soliton solutions. For the details, we refer readers to [1]}, [4]}.
